We study the “hyperboloidal Cauchy problem” for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behavior at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a $\lambda {\phi }^p$ nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal...